Recent developments of Mond-Pečarić method in operator inequalities. Inequalities for bounded selfadjoint operators on a Hilbert space. II
\textit{B. Mond} and \textit{J. E. Pečarić} [Houston J. Math. 20, No. 4, 645--651 (1994; Zbl 0819.47023)] established a method which gives a converse to Jensen's inequality associated with convex functions. Such a method can be used for finding the converse of other inequalities such as the Davis-Choi-Jensen inequality. This interesting book is to be regarded as the second volume of [\textit{T. Furuta} et al., Mond-Pečaric' method in operator inequalities. Inequalities for bounded selfadjoint operators on a Hilbert space. I. Zagreb: ELEMENT (2005; Zbl 1135.47012)], devoted to the Mond-Pečarić method in the field of self-adjoint operators on Hilbert spaces, and includes other results related to operator inequalities. The book consists of eleven chapters ended by some historical notes, an index, and a bibliography consisting of 295 items.NEWLINENEWLINEIn Chapter 1, the authors provide some basic facts about Hilbert space operators. Chapter 2 presents a history of the Kantorovich inequality as well as some ratio and difference type converses of operator versions of Jensen's inequality. There are some overlaps between the two books. One of them is the Furuta inequality and its proof in Chapter 3. This chapter also includes the Heinz inequality. Recently, several reverse Heinz inequalities have appeared in the literature which may be considered in the next volume of this series. In Chapter 4, the authors investigate the Kantorovich type inequalities related to the operator ordering and the chaotic one. Chapter 5 deals with converses of the Araki-Cordes inequality and the Ando-Hiai inequality. The geometric mean of \(n\) operators due to Ando-Li-Mathias and Lawson-Lim is investigated in Chapter 6. In this chapter, a converse of the weighted arithmetic-geometric mean inequality of \(n\) operators is presented. In Chapter 7, the differential-geometrical structure of operators is studied in details. Some properties of Mercer's type operator inequalities are established in Chapter 8. The next chapter deals with inequalities involving continuous fields of operators. A treatment of the Jensen operator inequality without operator convexity is given in Chapter 10, based on \textit{J. Mićić Hot} et al.\ [Linear Algebra Appl. 434, No. 5, 1228--1237 (2011; Zbl 1216.47026)]. The last chapter includes operator versions of the Bohr and Dunkl-Williams inequalities; see the surveys of \textit{M. Fujii} et al.\ [Springer Optimization and Its Applications 68, 279--290 (2012; Zbl 1262.47027)] and \textit{M. S. Moslehian} et al.\ [Banach J. Math. Anal. 5, No. 2, 138--151 (2011; Zbl 1225.47022)].NEWLINENEWLINE Some topics of the book are closely related to inequalities in the setting of Hilbert \(C^*\)-modules, which the reviewer hopes to be considered in another volume. The book is rather self-contained, but in some places the reader is better to be referred to the first volume, in particular with regard to operator means. Overall, the book presents an innovative and compelling perspective on operator inequalities and is enjoyable to read.
scientific article; zbMATH DE number 6183114